Matrix transformation inquiry

The prompt

Mathematical inquiry processes: Test cases; conjecture, generalise and prove. Conceptual field of inquiry: Transformation matrices; matrix multiplication.

The matrix multiplication prompt was designed for an A-level Further Mathematics class. The 12 students had already studied matrix arithmetic and how to find an inverse. In the orientation phase of the inquiry, the students posed questions about the prompt and discussed the conditions under which it might be true:

Preliminary inquiry

To answer the question about whether there are matrices that represent all the transformations, the teacher might choose to carry out a preliminary inquiry (see the PowerPoint in the Resources below). Students determine what each matrix represents. 

Note: Translations can be represented by matrix addition and subtraction. The teacher is advised to exclude translations from the main inquiry after an initial discussion or explanation of how the matrices work.

Testing cases

After the orientation phase, the class chose to test cases in order to verify a rule. Is the single matrix  that represents a combined transformation the product of AB or BA or could it be both? A few students spontaneously started to develop conjectures, propose generalisations and construct algebraic proofs. We present examples of their inquiries below.

True or false?

If we consider combinations of reflections, rotations (anti-clockwise about the origin), enlargements and shears (both with centres at the origin), then the statement in the prompt is always true. The single matrix in the case where matrix A represents the first transformation and matrix B the second is BA.

January 2023

Classroom inquiry

1. Test different cases

2. Conjecture and generalise

3. Prove

Conjecture and prove 

As his Further Maths class discussed the prompt, Qiang, a year 12 student, was thinking on his own. When it was his turn to feed back, he hesitantly put forward a conjecture: All transformation matrices are square in n dimensions. While his peers were exploring, Qiang set about proving his conjecture. He wrote up his notes (see below) for the next lesson and explained them to the class. 

In the final section of the notes, Qiang has drawn on the concept of extrusion from computer graphics. He explained that extrusion means to render a 2-dimensional object in three dimensions in a similar manner to turning a square into a cube.

February 2024